A boycicle is another term for a boy toy, or easily manipulated boyfriend. It is from the context of the boy being ridden like a bicycle. This sexual image, however, is extended as a metaphor to be emblematic of the relationship between the boy and girl (or homosexual boy) so that the girl is dominant throughout all aspects.
Unlike a boytoy, however, it is important to note that this stems from sexual experience whereas a boytoy is someone who is essentially manipulated through sex.
She slept with him like one night and now he is totally her boycycle.